{
  "id": "1602.03113",
  "version": "v1",
  "published": "2016-02-09T18:41:43.000Z",
  "updated": "2016-02-09T18:41:43.000Z",
  "title": "Rank 2 wall-crossing and the Serre correspondence",
  "authors": [
    "M. Kool",
    "A. Gholampour"
  ],
  "comment": "19 pages",
  "categories": [
    "math.AG",
    "hep-th"
  ],
  "abstract": "Let $\\mathcal{R}$ be a rank 2 reflexive sheaf on a smooth projective 3-fold $X$. We are interested in the Euler characteristics of the Quot schemes $\\mathrm{Quot}(\\mathcal{R},n)$ of 0-dimensional quotients of $\\mathcal{R}$ of length $n$. Provided $\\mathcal{R}$ admits a cosection cutting out a 1-dimensional subscheme, we prove that the generating function of these Euler characteristics is equal to $M(q)^{2e(X)}$ times a polynomial of degree $c_3(\\mathcal{R})$. This polynomial is the generating function of Euler characteristics of $\\mathrm{Quot}(\\mathcal{E}{\\it{xt}}^1(\\mathcal{R},\\mathcal{O}_X),n)$. Here $\\mathcal{E}{\\it{xt}}^1(\\mathcal{R},\\mathcal{O}_X)$ is a 0-dimensional sheaf supported at the points where $\\mathcal{R}$ is not locally free. Since $\\mathcal{R}$ is reflexive, it admits a 2-term resolution by vector bundles. In the case the vector bundles are of rank 1 and 3, $\\mathcal{E}{\\it{xt}}^1(\\mathcal{R},\\mathcal{O}_X)$ is a structure sheaf. This observation is used to prove a closed product formula for the Euler characteristics of $\\mathrm{Quot}(\\mathcal{R},n)$ in the case $X = \\mathbb{C}^3$ and $\\mathcal{R}$ is $T$-equivariant. This formula was first found using localization techniques and the double dimer model by the authors and B. Young. Our result follows from R. Hartshorne's Serre correspondence and a rank 2 version of the Hall algebra calculation of J. Stoppa and R. P. Thomas.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-09T18:41:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14C05",
      "14F05",
      "14H50",
      "14J30",
      "14N35"
    ],
    "keywords": [
      "euler characteristics",
      "vector bundles",
      "generating function",
      "hartshornes serre correspondence",
      "hall algebra calculation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160203113G",
      "inspire": 1420580
    }
  }
}